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\title{实变函数第五章：积分论}
\author{CQX ET AL}

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\begin{frame}{第五章目录 }

\begin{enumerate}

\item[5.1.] 黎曼积分的局限性、勒贝格积分简介
\item[5.2.] 非负简单函数的勒贝格积分
\item[5.3.] 非负可测函数的勒贝格积分
\item[5.4.] 一般可测函数的勒贝格积分
\item[5.5.] 黎曼积分和勒贝格积分
\item[5.6.] 勒贝格积分的几何意义、富比尼定理

\end{enumerate}

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%\begin{frame}{第五章重点 }
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\begin{frame}{5.1.1. 黎曼积分的局限性、勒贝格积分简介 }

\begin{itemize}

\item  {\color{red}问题：举例说明黎曼积分的缺陷。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{5.1.2.  }

\begin{itemize}

\item  {\color{red}问题：勒贝格积分的思路是什么？ }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{5.2.1. 非负简单函数的勒贝格积分 }

\begin{itemize}

\item  {\color{red}问题：设 $E\subseteq\mathbb{R}^n$ 是可测集，什么是 $E$ 上的一个非负简单函数？ }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{5.2.2.  }

\begin{itemize}

\item  {\color{red}问题：可测集 $E$ 上的非负简单函数的勒贝格积分是怎么定义的？ }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{5.2.3. 定理1 }

\begin{itemize}

\item  {\color{red}问题：设 $E$ 是可测集，设 $\varphi(x)$ 是 $E$ 上的一个非负简单函数。证明： }
\begin{enumerate}
\item  {\color{red} 对任意非负实数 $c$, 有 $$ \int_E c\varphi(x)dx = c\int_E \varphi(x)dx. $$ }
\item  {\color{red} 设 $A,B$ 是 $E$ 的两个不相交的可测子集，则 $$ \int_{A\cap B} \varphi(x)dx = \int_A \varphi(x)dx + \int_B \varphi(x)dx. $$ }
\item  {\color{red} 设 $\{A_n\}$ 是 $E$ 的一列可测子集，满足 $A_1\subset A_2\subset A_3\subset\cdots\subset A_n\subset\cdots$ 与 $\cup_{n=1}^{\infty} A_n = E$, 则有 
$$\lim\limits_{n\to\infty}\int_{A_n} \varphi(x)dx = \int_E \varphi(x)dx. $$ }
\end{enumerate}

%%\item  解答：


\end{itemize}

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\begin{frame}{5.2.4. 定理2 }

\begin{itemize}

\item  {\color{red}问题：设 $E$ 是可测集，设 $\varphi(x)$ 和 $\psi(x)$ 是 $E$ 上的非负简单函数。证明： }
\begin{enumerate}
\item  {\color{red} $$\int_E [\varphi(x)+\psi(x)]dx = \int_E \varphi(x)dx + \int_E \psi(x)dx. $$ }
\item  {\color{red} 对任意非负实数 $\alpha,\beta$, 有
$$\int_E [\alpha\varphi(x) + \beta\psi(x)]dx = \alpha\int_E \varphi(x)dx + \beta\int_E \psi(x)dx. $$
 }
\end{enumerate}

%%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{5.3.1. 非负可测函数的勒贝格积分 }

\begin{itemize}

\item  {\color{red}问题：可测集 $E$ 上的非负可测函数 $f(x)$ 的勒贝格积分是怎么定义的？
什么时候称 $f(x)$ 在 $E$ 上是勒贝格可积的？ }

%\item  解答：


\end{itemize}

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\begin{frame}{5.3.2. 定理1  }

\begin{itemize}

\item  {\color{red}问题：设 $E\subset\mathbb{R}^n$ 是可测集，设 $f(x)$ 是 $E$ 上的一个非负可测函数，证明： }
\begin{enumerate}
\item  {\color{red} 若 $m(E)=0$, 则 $$\int_Ef(x)dx=0.$$  }
\item  {\color{red} 若 $\int_Ef(x)dx=0,$ 则 $f(x)=0$ 在 $E$ 上几乎处处成立。 }
\item  {\color{red} 若 $\int_Ef(x)dx<\infty,$ 则 $0\le f(x)<\infty$ 在 $E$ 上几乎处处成立。 }
\item  {\color{red} 设 $A,B$ 是 $E$ 的两个不相交的可测子集，则 $$ \int_{A\cap B} \varphi(x)dx = \int_A \varphi(x)dx + \int_B \varphi(x)dx. $$ }
\end{enumerate}

%%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{5.3.3. 定理2  }

\begin{itemize}

\item  {\color{red}问题：设 $f(x)$ 与 $g(x)$ 是可测集 $E\subset\mathbb{R}^n$ 上的非负可测函数，证明： }
\begin{enumerate}

\item  {\color{red} 若 $f(x)\le g(x)$ 在 $E$ 上几乎处处成立，则 $$\int_Ef(x)dx \le \int_Eg(x).$$ 
这时，若 $g(x)$ 在 $E$ 上勒贝格可积，则 $f(x)$ 在 $E$ 上也勒贝格可积。  }

\item  {\color{red} 若 $f(x)= g(x)$ 在 $E$ 上几乎处处成立，则 $$\int_Ef(x)dx = \int_Eg(x).$$  
特别地，若 $f(x)$ 在 $E$ 上几乎处处为零，则 $$\int_Ef(x)dx = 0. $$ }

\end{enumerate}

%%\item  解答：


\end{itemize}

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\begin{frame}{5.3.4. 定理3 }

\begin{itemize}

\item  {\color{red}莱维定理：设 $E\subset\mathbb{R}^n$ 是可测集，设 $\{f_n(x)\}$ 是 $E$ 上一列非负可测函数，设对任意 $x\in E$, 对任意正整数 $n$, 都有 $f_n(x)\le f_{n+1}(x)$. 记 $f(x) = \lim\limits_{n\to\infty} f_n(x)$, 则有 
$$\lim\limits_{n\to\infty}\int_E f_n(x)dx = \int_E f(x)dx. $$ 
}

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{5.3.5. 定理4    }

\begin{itemize}

\item  {\color{red}问题：设 $E\subset\mathbb{R}^n$ 是可测集，设 $f(x)$ 和 $g(x)$ 是 $E$ 上的非负可测函数， 设 $\alpha$ 和 $\beta$ 都是非负实数，则有 
$$\int_E [\alpha f(x) + \beta g(x)]dx = \alpha\int_E f(x)dx + \beta\int_E g(x)dx. $$
}

%\item  解答：


\end{itemize}

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\begin{frame}{5.3.6. 定理5    }

\begin{itemize}

\item  {\color{red}逐项积分定理：设 $E\subset\mathbb{R}^n$ 是可测集，设 $\{f_n(x)\}$ 是 $E$ 上一列非负可测函数，则有
$$\int_E \left( \sum\limits_{n=1}^{\infty} f_n(x) \right)dx = \sum\limits_{n=1}^{\infty} \int_E f_n(x)dx. $$ 
 }

%\item  解答：


\end{itemize}

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\begin{frame}{5.3.7. 定理6    }

\begin{itemize}

\item  {\color{red}法图引理：设 $E\subset\mathbb{R}^n$ 是可测集，设 $\{f_n(x)\}$ 是 $E$ 上一列非负可测函数，则有
$$ \int_E \varliminf_{n\to\infty} f_n(x)dx \le \varliminf_{n\to\infty} \int_E f_n(x)dx. $$ 
 }

%\item  解答：


\end{itemize}

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\begin{frame}{5.3.8.  }

\begin{itemize}

\item  {\color{red}问题：举例说明法图引理中的小于等于不能改成等于。 }

%\item  解答：


\end{itemize}

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\begin{frame}{5.4.1. 一般可测函数的勒贝格积分  }

\begin{itemize}

\item  {\color{red}问题：设 $E\subset\mathbb{R}^n$ 是可测集，设 $f(x)$ 是 $E$ 上的可测函数，勒贝格积分 
$$\int_E f(x)dx$$  
是怎么定义的？什么时候称 $f(x)$ 在 $E$ 上是勒贝格可积的？ }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{5.4.2. 定理1  }

\begin{itemize}

\item  {\color{red}问题：设 $E\subset\mathbb{R}^n$ 是可测集，则有 }
\begin{enumerate}
\item  {\color{red} 若 $E\neq\varnothing$ 但 $m(E)=0$, 则 $E$ 上的任何实函数 $f$ 都是勒贝格可积的，且积分为零。  }
\item  {\color{red} 若 $f\in L(E)$, 则 $f(x)$ 在 $E$ 上是几乎处处有限的。 }
\item  {\color{red} 设 $f\in L(E)$, 设 $E=A\cup B$, 其中 $A,B$ 是 $E$ 的两个互不相交的可测集，则 $$\int_E f(x)dx = \int_A f(x)dx + \int_B  f(x)dx. $$  }

\end{enumerate}


\end{itemize}

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\begin{frame}{5.4.2. 定理1+  }

\begin{itemize}

\item  
\begin{enumerate}\setcounter{enumi}{3}%\itemsep0.5em

\item  {\color{red} 设 $f\in L(E)$, 设 $f(x)=g(x)$ 在 $E$ 上几乎处处成立，则 $g\in L(E)$, 且 $$\int_E f(x)dx = \int_E g(x)dx. $$ }

\item  {\color{red} 设 $f,g\in L(E)$, 设 $f(x)\le g(x)$ 在 $E$ 上几乎处处成立，则 
$$\int_E f(x)dx \le \int_E g(x)dx. $$ 
特别地，设 $m(E)<\infty$, 且 $b\le f(x)\le B$ 在 $E$ 上几乎处处成立，则 
$$bm(E) \le \int_E f(x)dx \le Bm(E). $$ }

\end{enumerate}


\end{itemize}

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\begin{frame}{5.4.2. 定理1++  }

\begin{itemize}

\item  
\begin{enumerate}\setcounter{enumi}{5}%\itemsep0.5em

\item  {\color{red} 设 $f\in L(E)$, 则 $|f|\in L(E)$, 且有  $$\left\vert \int_E f(x)dx \right\vert \le \int_E \left\vert f(x) \right\vert dx. $$ }

\item  {\color{red}  设 $f$ 是 $E$ 上的可测函数，设 $g$ 是 $E$ 上的非负勒贝格可积函数，且 $|f(x)|\le g(x)$ 在 $E$ 上几乎处处成立，则 
$$\left\vert \int_E f(x)dx \right\vert \le \int_E \left\vert f(x) \right\vert dx \le \int_E g(x)dx. $$ }

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.4.3. 定理2  }

\begin{itemize}

\item  {\color{red}问题：设 $E\subset\mathbb{R}^n$ 是可测集，设 $f,g\in L(E)$, 则有 }
\begin{enumerate}
\item  {\color{red} 对任意实数 $\lambda$, 有 $\lambda f\in L(E)$, 且 $$\int_E \lambda f(x)dx = \lambda \int_E f(x)dx. $$ }
\item  {\color{red} 有 $f+g\in L(E)$, 且 $$\int_E [ f(x) + g(x)]dx = \int_E f(x)dx + \int_E g(x)dx. $$ }
\item  {\color{red} 对任意实数 $\alpha,\beta$, 有 $\alpha f + \beta g\in L(E)$, 且 $$\int_E [\alpha f(x) + \beta g(x)]dx = \alpha\int_E f(x)dx + \beta\int_E g(x)dx. $$ }

\end{enumerate}

%%\item  解答：


\end{itemize}

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\begin{frame}{5.4.4. 定理3  }

\begin{itemize}

\item  {\color{red}积分的绝对连续性：设 $E\subset\mathbb{R}^n$ 是可测集，设 $f\in L(E)$, 则对任意 $\varepsilon>0$, 存在 $\delta>0$, 使得对任意可测集 $A\subset E$, 只要 $m(A)<\delta$, 就有
$$\left\vert \int_A f(x)dx \right\vert \le \int_A \left\vert f(x) \right\vert dx<\varepsilon. $$ }

%%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{5.4.5. 定理4  }

\begin{itemize}

\item  {\color{red}积分的可数可加性：设 $E\subset\mathbb{R}^n$ 是可测集，设 $E=\cup_{n=1}^{\infty} E_n$, 
这里每个 $E_n$ 都是可测集，且互不相交。设 $f$ 在 $E$ 上的积分确定，则 
$$ \int_E f(x)dx =\sum\limits_{n=1}^{\infty} \int_{E_n} f(x)  dx. $$ 
}

%%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{5.4.6. 定理5  }

\begin{itemize}

\item  {\color{red}勒贝格控制收敛定理：设 $E\subset\mathbb{R}^n$ 是可测集，设 $\{f_n(x)\}$ 是 $E$ 上的一列可测函数，设 $F(x)$ 是 $E$ 上的非负勒贝格可积函数。设对任意的正整数 $n$, $|f_n(x)|\le F(x)$ 在 $E$ 上几乎处处成立，且 $\lim\limits_{n\to\infty} f_n(x)=f(x)$ 在 $E$ 上几乎处处成立，则
\begin{eqnarray*}
\lim\limits_{n\to\infty} \int_E |f_n(x)-f(x)|dx &=& 0, \\ 
\lim\limits_{n\to\infty} \int_E f_n(x) dx &=& \int_E f(x) dx. 
\end{eqnarray*}
}

%%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{5.4.7. 定理6  }

\begin{itemize}

\item  {\color{red}问题：设 $E\subset\mathbb{R}^n$ 是可测集，设 $\{f_n(x)\}$ 与 $f(x)$ 都是 $E$ 上的可测函数，
设 $F(x)$ 是 $E$ 上的非负勒贝格可积函数。
设对任意的正整数 $n$, $|f_n(x)|\le F(x)$ 在 $E$ 上几乎处处成立，且 $f_n(x)$ 在 $E$ 上依测度收敛于 $f(x)$, 则
\begin{eqnarray*}
\lim\limits_{n\to\infty} \int_E |f_n(x)-f(x)|dx &=& 0, \\ 
\lim\limits_{n\to\infty} \int_E f_n(x) dx &=& \int_E f(x) dx. 
\end{eqnarray*}
 }

%%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{5.4.8. 定理7  }

\begin{itemize}

\item  {\color{red}问题：设 $E\subset\mathbb{R}^n$ 是可测集，设 $\{f_n(x)\}$ 是 $E$ 上的一列勒贝格可积函数。
设正项级数 $$\sum\limits_{n=1}^{\infty} \int_E |f_n(x)|dx$$ 收敛，
则函数项级数 $$\sum\limits_{n=1}^{\infty} f_n(x)$$ 在 $E$ 上几乎处处收敛，
其和函数 $F(x)$ 在 $E$ 上勒贝格可积，且 
$$ \int_E \left( \sum\limits_{n=1}^{\infty} f_n(x) \right) dx =\sum\limits_{n=1}^{\infty} \int_E f_n(x) dx. $$ 
}

%%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{5.4.9. 定理8  }

\begin{itemize}

\item  {\color{red}问题：设 $E\subset\mathbb{R}^n$ 是可测集，设 $f(x,t)$ 是 $E\times (a,b)$ 上的实函数。
设对任意的 $t\in (a,b)$, 作为 $x$ 的函数 $f(x,t)$ 在 $E$ 上是勒贝格可积的，
设对任意的 $x\in E$, 作为 $t$ 的函数 $f(x,t)$ 在 $(a,b)$ 上是可导的，且 
$$\left\vert \frac{\partial }{\partial t}(f(x,t)) \right\vert \le F(x), $$
这里 $F(x)$ 是 $E$ 上的某个非负勒贝格可积函数。则 $\int_E f(x,t)dx$ 作为 $t$ 的函数在 $(a,b)$ 种可导，且
$$\frac{d}{dt} \int_E f(x,t)dx = \int_E \frac{\partial }{\partial t}(f(x,t)) dx. $$
 }

%%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{5.4.10. 例子  }

\begin{itemize}

\item  {\color{red}问题：设 $f\in L[a,b]$, 则对任意 $\varepsilon>0$, 存在 $g\in C[a,b]$, 使得
$$\int_{[a,b]} |f(x)-g(x)|dx < \varepsilon. $$
 }

%%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{5.5.1. 黎曼积分和勒贝格积分  }

\begin{itemize}

\item  {\color{red}问题：勒贝格积分是黎曼积分的推广，但不是黎曼反常积分的推广。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{5.5.2. 定理1 }

\begin{itemize}

\item  {\color{red}问题：设 $f(x)$ 是 $[a,b]$ 上的有界函数，则 $f(x)$ 在 $[a,b]$ 上黎曼可积的充分必要条件是 $f(x)$ 在 $[a,b]$ 上几乎处处连续，即不连续的点的全体组成一个零测度集。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{5.5.3. 定理2 }

\begin{itemize}

\item  {\color{red}问题：设 $f(x)$ 是 $[a,b]$ 上的一个有界函数，设 $f(x)$ 在 $[a,b]$ 上是黎曼可积的。则 $f(x)$ 在 $[a,b]$ 上是勒贝格可积的，且 $$(L)\int _{[a,b]} f(x)dx = (R) \int_a^b f(x)dx. $$ }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{5.5.4. 定理3 }

\begin{itemize}

\item  {\color{red}问题：设 $f(x)$ 是 $[a,\infty)$ 上的一个非负实函数。
设对任意的 $A>a$, 函数 $f(x)$ 在 $[a,A]$ 上是黎曼可积的，且黎曼反常积分 $$(R)\int_a^{\infty} f(x)dx$$ 收敛，
则 $f(x)$ 在 $[a,\infty)$ 上是勒贝格可积的，且 
$$(L)\int _{[a,\infty]} f(x)dx = (R) \int_a^\infty f(x)dx. $$
}

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{5.5.5. 例子 }

\begin{itemize}

\item  {\color{red}问题：设 $$f(x) = \left\{
\begin{array}{ll}
\frac{\sin x}{x}, & x>0, \\ 
1, & x=0,
\end{array}
\right. $$ 
则 $f(x)$ 在 $[0,\infty)$ 上连续，黎曼反常积分存在，但不是勒贝格可积的。
}

%\item  解答：


\end{itemize}

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\begin{frame}{5.6.1. 勒贝格积分的几何意义、富比尼定理 }

\begin{itemize}

\item  {\color{red}问题：什么是 $\mathbb{R}^n$ 中的点集的直积 $A\times B$?  什么是 $\mathbb{R}^n$ 中的点集的截面？}

%\item  解答：


\end{itemize}

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\begin{frame}{5.6.2. 定理1 }

\begin{itemize}

\item  {\color{red}截面定理：设 $E\subset \mathbb{R}^{p+q}$ 是可测集，则 }
\begin{enumerate}
\item  {\color{red} 对于 $\mathbb{R}^p$ 中的几乎所有的点 $x$, 截面 $E_x$ 是 $\mathbb{R}^q$ 中的可测集。  }
\item  {\color{red} 测度 $m(E_x)$ 作为 $x$ 的函数，是 $\mathbb{R}^p$ 上几乎处处有定义的可测函数。 }
\item  {\color{red}  $$m(E) = \int_{\mathbb{R}^p} m(E_x)dx. $$ }
\end{enumerate} 

%\item  解答：

\end{itemize}

\end{frame}

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\begin{frame}{5.6.3. 定理2 }

\begin{itemize}

\item  {\color{red}问题：设 $A,B$ 分别是 $\mathbb{R}^p, \mathbb{R}^q$ 中的可测集，则 $A\times B$ 是 $\mathbb{R}^{p+q}$ 中的可测集，且 $$m(A\times B) = m(A)m(B).$$ }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{5.6.4. 定理3: 非负可测函数的积分的几何意义 }

\begin{itemize}

\item  {\color{red}问题：设 $f(x)$ 是可测集 $E\subset \mathbb{R}^n$ 上的非负函数，
记 $$G(E,f) = \{(x,z): x\in E, 0\le z< f(x)\},$$ 
称为 $f(x)$ 在 $E$ 上的下方图形，则 }
\begin{enumerate}
\item  {\color{red} $f(x)$ 是 $E$ 上的可测函数的充分必要条件是 $G(E,f)$ 是 $\mathbb{R}^{n+1}$ 中的可测集。  }
\item  {\color{red} 当 $f(x)$ 在 $E$ 上可测时，$$\int_E f(x)dx = m(G(E,f)). $$  }

\end{enumerate} 



%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{5.6.5. 定理4: 富比尼定理 }

\begin{enumerate}

\item  {\color{red} 设 $f(P)=f(x,y)$ 在可测集 $A\times B\subset \mathbb{R}^{p+q}$ 上非负可测，
则对几乎处处 $x\in A$, $f(x,y)$ 作为 $y$ 的函数在 $B$ 上可测，且 
$$\int_{A\times B} f(P)dP = \int_Adx \int_B f(x,y)dy. $$ }

\item  {\color{red} 设 $f(P)=f(x,y)$ 在 $A\times B\subset \mathbb{R}^{p+q}$ 上可积，
则对几乎处处 $x\in A$, $f(x,y)$ 作为 $y$ 的函数在 $B$ 上可积，
$\int_B f(x,y)dy$ 作为 $x$ 的函数在 $A$ 上可积，且 
$$\int_{A\times B} f(P)dP = \int_Adx \int_B f(x,y)dy. $$ }

%\item  解答：


\end{enumerate}

\end{frame}

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\begin{frame}{5.6.6. 例子 }

\begin{itemize}

\item  {\color{red}问题：说明 $$f(x,y)=\frac{x^2-y^2}{(x^2+y^2)^2}$$ 在 $E=(0,1)\times (0,1)$ 上不是勒贝格可积的。 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{习题1  }

\begin{itemize}

\item  {\color{red}问题：设在康托尔集 $P$ 上定义函数 $f(x)=0$, 而在 $P$ 的余集中长为 $3^{-n}$ 的构成区间上定义为 
$n$, $n=1,2,\cdots$, 证明 $f(x)$ 可积，并求出积分值。
}

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{习题7  }

\begin{itemize}

\item  {\color{red}问题：设 $$f(x) = \frac{\sin(1/x)}{x^\alpha},\,\, 0<x\le 1,$$ 
讨论 $\alpha$ 为何值时，$f(x)$ 在 $(0,1]$ 上是勒贝格可积的。
}

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{习题12  }

\begin{itemize}

\item  {\color{red}问题：试从 $\frac{1}{1+x} = (1-x) + (x^2-x^3) + \cdots (0<x<1)$ 证明
$$\ln 2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots $$
 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{习题13 }

\begin{itemize}

\item  {\color{red}问题：证明：
$$\lim\limits_{n\to\infty} \int_0^\infty \frac{dt}{\left( 1+\frac{t}{n} \right)^n t^{\frac{1}{n}}} =1. $$
 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{习题14  }

\begin{itemize}

\item  {\color{red}问题：若 $p>-1$, 证明：
$$\int_0^1 \frac{x^p}{1-x}\ln\frac{1}{x}dx =\sum\limits_{n=1}^{\infty} \frac{1}{(p+n)^2}. $$
 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{习题24  }

\begin{itemize}

\item  {\color{red}问题：设 $\{r_k\}$ 是 $[0,1]$ 中的全体有理数，证明：
$$\sum\limits_{k=1}^{\infty} \frac{1}{k^2\sqrt{|x-r_k|}}$$
在 $[0,1]$ 上几乎处处收敛。
 }

%\item  解答：


\end{itemize}

\end{frame}

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\begin{frame}{习题25  }

\begin{itemize}

\item  {\color{red}问题：设 $f(x)$ 是 $\mathbb{R}$ 上的勒贝格可积函数，证明：
$$\sum\limits_{n=1}^{\infty} f(x+n)$$
在 $\mathbb{R}$ 上几乎处处绝对收敛。
}

%\item  解答：


\end{itemize}

\end{frame}



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\end{document}



